cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A005160 Number of alternating sign n X n matrices invariant under a quarter turn.

Original entry on oeis.org

1, 0, 1, 2, 3, 0, 12, 40, 100, 0, 1225, 6860, 28812, 0, 1037232, 9779616
Offset: 1

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Author

Keywords

Comments

Robbins incorrectly gives a(12) = 6460.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

Crossrefs

a(4n) gives A059476.

Formula

Robbins gives a simple (conjectured) formula.

A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's).

Original entry on oeis.org

1, 1, 1, 2, 6, 33, 286, 4420, 109820, 4799134, 340879665, 42235307100, 8564558139000, 3012862604463000, 1742901718473961200, 1742218029490675762080, 2873822682985675809192288, 8167157387273280570395662320, 38402596062535617548517706584760, 310388509293255836481583597538626504
Offset: 0

Views

Author

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

Crossrefs

Programs

  • PARI
    \\ here b(n) and c(n) are A005156 and A051255.
    b(n) = prod(k=0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
    c(n) = prod(k=0, n-1, (3*k+1)*(6*k)!*(2*k)!/((4*k)!*(4*k+1)!));
    a(n) = b(n\2) * c((n+1)\2) \\ Andrew Howroyd, May 09 2023

Formula

Robbins gives a simple (conjectured) formula, which was proven by Okada.
a(2*n) = A005156(n)*A051255(n); a(2*n+1) = A005156(n)*A051255(n+1). - Paul Zinn-Justin, May 05 2023
a(n) = A005156(floor(n/2)) * A051255(ceiling(n/2)). - Andrew Howroyd, May 09 2023

Extensions

More terms (from the P. Pyatov paper) from Vladeta Jovovic, Aug 15 2008
Terms a(13) and beyond from Andrew Howroyd, May 09 2023
Showing 1-2 of 2 results.